1. **State the problem:** Simplify the algebraic expression $$\frac{5}{8}x + \frac{3}{4}y + \frac{1}{2}x - \frac{1}{4}y$$. 2. **Identify like terms:** Terms with $x$ are $$\frac{5}{8}x$$ and $$\frac{1}{2}x$$. Terms with $y$ are $$\frac{3}{4}y$$ and $$-\frac{1}{4}y$$. 3. **Add the $x$ terms:** $$\frac{5}{8}x + \frac{1}{2}x = \frac{5}{8}x + \frac{4}{8}x = \left(\frac{5}{8} + \frac{4}{8}\right)x = \frac{9}{8}x$$. 4. **Add the $y$ terms:** $$\frac{3}{4}y - \frac{1}{4}y = \left(\frac{3}{4} - \frac{1}{4}\right)y = \frac{2}{4}y = \frac{1}{2}y$$. 5. **Write the simplified expression:** $$\frac{9}{8}x + \frac{1}{2}y$$. **Final answer:** $$\frac{9}{8}x + \frac{1}{2}y$$